Geometry and Topology of Knots and Links

自20世纪80年代初以来，人们就知道纽结补满足几何化猜想。 因此，纽结补数的几何学应该是研究纽结的一个有用的工具。 然而，一般不知道的是如何将纽结的图与它们的几何结构联系起来，特别是双曲纽结。 在过去的几年里，PI已经发现了基于图的结补的某些几何信息的界限，包括结类的尖点形状和体积的界限。在这个项目中，PI将把过去几年发展起来的三维流形理论中的技术应用于纽结补，以进一步理解它们的几何性质。 在数学中，这样的环被称为结。 结理论的一个目标是根据结的快照（一个图）来确定它是否可以在不破坏环的情况下解开。 研究纽结的一种方法是不考虑纽结本身，而是考虑从三维球面中移除纽结所获得的三维空间，称为纽结补。 PI将利用三维空间或三维流形理论研究的最新进展来研究结。 这项研究在弦理论和DNA打结等领域都有应用。

It has been known since the early 1980's that knot complements satisfy the geometrization conjecture. Thus the geometry of knot complements ought to be a useful tool in the study of knots. However, what is not known in general is how to relate diagrams of knots to their geometric structure, particularly hyperbolic knots. In the last few years, the PI has found bounds on certain geometric information of a knot complement based on a diagram, including bounds on cusp shapes and volumes for classes of knots. In this project, the PI will apply techniques in 3-manifold theory developed in the last few years to knot complements, to further our understanding of their geometric properties.A closed loop lying in space may be knotted or unknotted. In mathematics, such a loop is called a knot. One goal of knot theory is to determine, based on a snapshot of that knot (a diagram), whether it can be unknotted without breaking the loop. A method of studying knots is to consider not the knot itself, but the 3-dimensional space obtained by removing the knot from the 3-sphere, called the knot complement. The PI will use recent advances in the study of 3-dimensional spaces, or 3-manifold theory, to study knots. This research has applications to such areas as string theory and the knotting of DNA.